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Tracking Processes with Change Points

  • Shelemyahu Zacks
Chapter
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Part of the Statistics for Industry, Technology, and Engineering book series (SITE)

Abstract

In Sect.  1.2 of the Introduction we described a problem of tracking a missile in order to quickly detect significant deviation from its designed orbit. The objective is to estimate the expected value in a given time of a random process under observation, when the data available is only the past observations. The present chapter is based on the papers listed among the references. We start with the joint paper of Chernoff and Zacks (Ann Math Stat 35:999–1018, 1964). For simplification, we assume that the orbit is a given horizontal line at level μ. The observations are taken at discrete time points, and they reflect the random positions of the object (missile) around μ, and the possible changes in the horizontal trend. We assume here that as long as there are no changes in the trend, the observations are realization of independent random variables, having a common normal distribution, \(N(\mu ,\sigma _{\epsilon }^{2})\). Since the objective is to detect whether changes have occurred in the trend, we let μn = E{Xn}, where {X1, …, Xn} is the given data, and adopt the following model
$$\displaystyle \begin{aligned} X_{n} =\mu _{1} +\sum _{i =1}^{n -1}J_{n -i}Z_{n -i} +\epsilon _{n},n \geq 1. \end{aligned}$$

References

  1. Bather, A. J. (1963). Control charts and minimization of costs. Journal of the Royal Statistical Society, B, 25, 49–80.zbMATHGoogle Scholar
  2. Bather, J. A. (1967). On a quickest detection problem. Annals of Mathematical Statistics, 38, 711–724.MathSciNetCrossRefGoogle Scholar
  3. Brown, M., & Zacks, S. (2006). A note on optimal stopping for possible change in the intensity of an ordinary Poisson process. Statistics & Probability Letters, 76, 1417–1425.MathSciNetCrossRefGoogle Scholar
  4. Chernoff, H., & Zacks, S. (1964). Estimating the current mean of a normal distribution which is subject to changes in time. Annals of Mathematical Statistics, 35, 999–1018.MathSciNetCrossRefGoogle Scholar
  5. Johnson, N. L., & Kotz, S. (1969). Distributions in statistics: Discrete distributions. Boston: Houghton and Mifflin.zbMATHGoogle Scholar
  6. Kenett, R. S., & Zacks, S. (2014). Modern industrial statistics with R, minitab and JMP (2nd ed.). Chichester: Wiley.zbMATHGoogle Scholar
  7. Lorden, G. (1971). Procedures for reacting to a change in distribution. The Annals of Mathematical Statistics, 42, 1897–190.MathSciNetCrossRefGoogle Scholar
  8. Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41, 100–114.MathSciNetCrossRefGoogle Scholar
  9. Page, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika, 42, 523–527.MathSciNetCrossRefGoogle Scholar
  10. Page, E. E. (1957). On problems in which a change of parameter occurs at an unknown point. Biometrika, 44, 248–252.CrossRefGoogle Scholar
  11. Peskir, G., & Shiryaev, A. N. (2002). Solving the Poisson disorder problem. In K. Sandman & P. J. Schonbucher (Eds.). Advances in finance and stochastics (pp. 295–312). New York: Springer.CrossRefGoogle Scholar
  12. Plunchenko, A. S., & Tartakovsky, A. G. (2010). On optimality of Shiryaev-Roberts procedure for detecting a change in distributions. The Annals of Statistics, 39, 3445–3457.MathSciNetCrossRefGoogle Scholar
  13. Pollak, M. (1985). Optimal detection of a change in distribution. The Annals of Statistics, 15, 749–779.MathSciNetCrossRefGoogle Scholar
  14. Shewhart, W. A. (1931). Economic control of quality of manufactured product. Princeton: Van Nostrand.Google Scholar
  15. Shiryaev, A. N. (1963). On optimum methods in quickest detection problem. Theory of Probability and its Applications, 7, 22–46.CrossRefGoogle Scholar
  16. Zacks, S. (1983). Survey of classical and Bayesian approaches to the change point problem: Fixed sample and sequential procedures of testing and estimation. In D. Siegmund, J. Rustagi, & G. Gaseb Rizvi (Eds.). Recent advances in statistics (pp. 245–269). New York: Academic.CrossRefGoogle Scholar
  17. Zacks S. (1985). Distribution of stopping variables in sequential procedures for the detection of shifts in the distributions of discrete random variables. Communications in Statistics, B9, 1–8.Google Scholar
  18. Zacks, S. (2004). Distribution of failure times associated with non-homogeneous compound Poisson damage processes. IMS Lecture Notes, 45, 396–407.MathSciNetzbMATHGoogle Scholar
  19. Zacks, S. (2009). Stage-wise adaptive designs. New York: Wiley.CrossRefGoogle Scholar
  20. Zacks, S., & Barzily, Z. (1981). Bayes procedures for detecting a shift in the probability of success in a series of Bernoulli trials. Journal of Statistics Planning & Inference, 5, 107–119.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Shelemyahu Zacks
    • 1
  1. 1.McLeanUSA

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